Doctor of Philosophy

dissertationPartial differential equations (PDEs) are widely used in science and engineering to model phenomena such as sound, heat, and electrostatics. In many practical science and engineering applications, the solutions of PDEs require the tessellation of computational domains into unstructured meshes and entail computationally expensive and time-consuming processes. Therefore, efficient and fast PDE solving techniques on unstructured meshes are important in these applications. Relative to CPUs, the faster growth curves in the speed and greater power efficiency of the SIMD streaming processors, such as GPUs, have gained them an increasingly important role in the high-performance computing area. Combining suitable parallel algorithms and these streaming processors, we can develop very efficient numerical solvers of PDEs. The contributions of this dissertation are twofold: proposal of two general strategies to design efficient PDE solvers on GPUs and the specific applications of these strategies to solve different types of PDEs. Specifically, this dissertation consists of four parts. First, we describe the general strategies, the domain decomposition strategy and the hybrid gathering strategy. Next, we introduce a parallel algorithm for solving the eikonal equation on fully unstructured meshes efficiently. Third, we present the algorithms and data structures necessary to move the entire FEM pipeline to the GPU. Fourth, we propose a parallel algorithm for solving the levelset equation on fully unstructured 2D or 3D meshes or manifolds. This algorithm combines a narrowband scheme with domain decomposition for efficient levelset equation solving

Diffusion Reaction Eikonal Alternant Model: Towards Fast Simulations of Complex Cardiac Arrhythmias

Reaction-diffusion (RD) computer models are suitable to investigate the mechanisms of cardiac arrthymias but not directly applicable in clinical settings due to their computational cost. On the other hand, alternative faster eikonal models are incapable of reproducing reentrant activation when solved by iterative methods. The diffusion reaction eikonal alternant model (DREAM) is a new method in which eikonal and RD models are alternated to allow for reactivation. To solve the eikonal equation, the fast iterative method was modified and embedded into DREAM. Obtained activation times control transmembrane voltage courses in the RD model computing, while repolarization times are provided back to the eikonal model. For a planar wave-front in the center of a 2D patch, DREAM action potentials (APs) have a small overshoot in the upstroke compared to pure RD simulations (monodomain) but similar AP duration. DREAM conduction velocity does not increase near boundaries or stimulated areas as it occurs in RD. Anatomical reentry was reproduced with the S1-S2 protocol. This is the first time that an iterative method is used to solve the eikonal model in a version that admits reactivation. This method can facilitate uptake of computer models in clinical settings. Further improvements will allow to accurately represent even more complex patterns of arrhythmi

Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the $L^1$ norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. Moreover, the implementation of parallel computing using domain decomposition with $1$-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code

Comparison of Propagation Models and Forward Calculation Methods on Cellular, Tissue and Organ Scale Atrial Electrophysiology

The bidomain model and the finite element method are an established standard to mathematically describe cardiac electrophysiology, but are both suboptimal choices for fast and large-scale simulations due to high computational costs. We investigate to what extent simplified approaches for propagation models (monodomain, reaction-Eikonal and Eikonal) and forward calculation (boundary element and infinite volume conductor) deliver markedly accelerated, yet physiologically accurate simulation results in atrial electrophysiology. Methods: We compared action potential durations, local activation times (LATs), and electrocardiograms (ECGs) for sinus rhythm simulations on healthy and fibrotically infiltrated atrial models. Results: All simplified model solutions yielded LATs and P waves in accurate accordance with the bidomain results. Only for the Eikonal model with pre-computed action potential templates shifted in time to derive transmembrane voltages, repolarization behavior notably deviated from the bidomain results. ECGs calculated with the boundary element method were characterized by correlation coefficients >0.9 compared to the finite element method. The infinite volume conductor method led to lower correlation coefficients caused predominantly by systematic overestimations of P wave amplitudes in the precordial leads. Conclusion: Our results demonstrate that the Eikonal model yields accurate LATs and combined with the boundary element method precise ECGs compared to markedly more expensive full bidomain simulations. However, for an accurate representation of atrial repolarization dynamics, diffusion terms must be accounted for in simplified models. Significance: Simulations of atrial LATs and ECGs can be notably accelerated to clinically feasible time frames at high accuracy by resorting to the Eikonal and boundary element methods

Geodesics in Heat

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

Digital twinning of cardiac electrophysiology models from the surface ECG: a geodesic backpropagation approach

The eikonal equation has become an indispensable tool for modeling cardiac electrical activation accurately and efficiently. In principle, by matching clinically recorded and eikonal-based electrocardiograms (ECGs), it is possible to build patient-specific models of cardiac electrophysiology in a purely non-invasive manner. Nonetheless, the fitting procedure remains a challenging task. The present study introduces a novel method, Geodesic-BP, to solve the inverse eikonal problem. Geodesic-BP is well-suited for GPU-accelerated machine learning frameworks, allowing us to optimize the parameters of the eikonal equation to reproduce a given ECG. We show that Geodesic-BP can reconstruct a simulated cardiac activation with high accuracy in a synthetic test case, even in the presence of modeling inaccuracies. Furthermore, we apply our algorithm to a publicly available dataset of a rabbit model, with very positive results. Given the future shift towards personalized medicine, Geodesic-BP has the potential to help in future functionalizations of cardiac models meeting clinical time constraints while maintaining the physiological accuracy of state-of-the-art cardiac models.Comment: 9 pages, 5 figure

Novel Methods to Incorporate Physiological Prior Knowledge into the Inverse Problem of Electrocardiography - Application to Localization of Ventricular Excitation Origins

17 Millionen Todesfälle jedes Jahr werden auf kardiovaskuläre Erkankungen zurückgeführt. Plötzlicher Herztod tritt bei ca. 25% der Patienten mit kardiovaskulären Erkrankungen auf und kann mit ventrikulärer Tachykardie in Verbindung gebracht werden. Ein wichtiger Schritt für die Behandlung von ventrikulärer Tachykardie ist die Detektion sogenannter Exit-Points, d.h. des räumlichen Ursprungs der Erregung. Da dieser Prozess sehr zeitaufwändig ist und nur von fähigen Kardiologen durchgeführt werden kann, gibt es eine Notwendigkeit für assistierende Lokalisationsmöglichkeiten, idealerweise automatisch und nichtinvasiv. Elektrokardiographische Bildgebung versucht, diesen klinischen Anforderungen zu genügen, indem die elektrische Aktivität des Herzens aus Messungen der Potentiale auf der Körperoberfläche rekonstruiert wird. Die resultierenden Informationen können verwendet werden, um den Erregungsursprung zu detektieren. Aktuelle Methoden um das inverse Problem zu lösen weisen jedoch entweder eine geringe Genauigkeit oder Robustheit auf, was ihren klinischen Nutzen einschränkt. Diese Arbeit analysiert zunächst das Vorwärtsproblem im Zusammenhang mit zwei Quellmodellen: Transmembranspannungen und extrazelluläre Potentiale. Die mathematischen Eigenschaften der Relation zwischen den Quellen des Herzens und der Körperoberflächenpotentiale werden systematisch analysiert und der Einfluss auf das inverse Problem verdeutlicht. Dieses Wissen wird anschließend zur Lösung des inversen Problems genutzt. Hierzu werden drei neue Methoden eingeführt: eine verzögerungsbasierte Regularisierung, eine Methode basierend auf einer Regression von Körperoberflächenpotentialen und eine Deep-Learning-basierte Lokalisierungsmethode. Diese drei Methoden werden in einem simulierten und zwei klinischen Setups vier etablierten Methoden gegenübergestellt und bewertet. Auf dem simulierten Datensatz und auf einem der beiden klinischen Datensätze erzielte eine der neuen Methoden bessere Ergebnisse als die konventionellen Ansätze, während Tikhonov-Regularisierung auf dem verbleibenden klinischen Datensatz die besten Ergebnisse erzielte. Potentielle Ursachen für diese Ergebnisse werden diskutiert und mit Eigenschaften des Vorwärtsproblems in Verbindung gebracht